We present an extension of the protector control scheme introduced for the linear case in a previous work to a class of nonlinear systems. The systems considered are assumed to have a finite propagation velocity while the initial state is subject to a spreading disturbance. We characterize such a control first by using the remediability approach to the resulting nonlinear delay system, and then by coupling families of transformations and the delay approach. To illustrate this work, we provide a simulation example.
@article{bwmeta1.element.bwnjournal-article-amcv20i3p427bwm, author = {Youssef Qaraai and Abdes Samed Bernoussi}, title = {Protector control: Extension to a class of nonlinear distributed systems}, journal = {International Journal of Applied Mathematics and Computer Science}, volume = {20}, year = {2010}, pages = {427-443}, zbl = {1211.93073}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i3p427bwm} }
Youssef Qaraai; Abdes Samed Bernoussi. Protector control: Extension to a class of nonlinear distributed systems. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 427-443. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i3p427bwm/
[000] Afifi, L., Chafiai, A. and El Jai, A. (2002). Regionally efficient and strategic actuators, International Journal of Systems Science 33(4): 1-12. | Zbl 1013.93029
[001] Afifi, L., El Jai, A. and Merry, M. (2000). Detection and sources reconstruction in a tube, International Journal of Systems Science 31(2): 149-159.
[002] Afifi, L., El Jai, A. and Merry, M. (2001). Regional detection and reconstruction of unknown internal or boundary sources, International Journal of Applied Mathematics and Computer Science 11(2): 319-348. | Zbl 0979.93052
[003] Beltrami, E. (1987). Mathematics for Dynamic Modelling, Academic Press, San Diego, CA. | Zbl 0625.58012
[004] Bernoussi, A. (2007). Spreadability and vulnerability of distributed parameter systems, International Journal of Systems Science 38(4): 305-317. | Zbl 1115.93007
[005] Bernoussi, A. and Amharref, M. (2003). Etalabilité-vulnrabilité, Annals of University of Craiova, Mathematics and Computer Science Series 30(4): 53-62.
[006] Bernoussi, A. and El Jai, A. (2000). New approach of spreadability, Journal of Mathematical and Computer Modelling 31(13): 93-109. | Zbl 0960.93018
[007] Bernoussi, A., El Jai, A. and Pritchard, A. J. (2001). Spreadability and evolving interfaces, International Journal of Systems Science 32(10): 1217-1232. | Zbl 1014.93015
[008] Bernoussi, A. (2010). Spreadability, vulnerability and protector control, Mathematical Modelling of Natural Phenomena 5(7): 145-150, DOI: 10.1051/mmnp/20105724.
[009] Curtain, R.F. and Pritchard, A.J. (1978). Infinite Dimensional Linear Systems Theory, Springer, Berlin. | Zbl 0389.93001
[010] Dautray, R.F. and Lions, J.. (1984). Analyse mathématique et calcul numérique pour les sciences et les techniques, Série Scientifique, Tome 3, Masson, Paris. | Zbl 0642.35001
[011] Diaz, J.I. and Lions, J. L. (1993). Mathematics Climate and Environment, Research in Applied Mathematics, RMA 27, Masson, Paris. | Zbl 0782.00023
[012] El Jai, A. (2002). Analyse régionale des systémes distribuès, SMAI (Société de Mathématiques Appliquées et Industrielles), ESAIM-European Series in Applied and Industrial Mathematics 8: 663-692.
[013] El Jai, A. (2004). Eléments d'Analyse et Contrôle des Systèmes, Collection études, University of Perpignan Press, Perpignan.
[014] El Jai, A. and Kassara, K. (1994). Spreadable distributed systems, Mathematical and Computer Modelling 20(1): 47-64. | Zbl 0829.92018
[015] El Jai, A. and Kassara, K. (1996). Spreadability of transport systems, International Journal of Systems Science 27(7): 681-688. | Zbl 0857.35125
[016] El Jai, A. and Pritchard, A.J. (1986). Capteurs et actionneurs dans l'analyse des systems distribués, RMA 3, Masson, Paris. | Zbl 0627.93001
[017] El Jai, A., Simon, M., Zerrik, E. and Pritchard, A.J. (1995). Regional controllability of distributed systems, International Journal of Control 62(6): 1351-1365. | Zbl 0844.93016
[018] Giuggioli, L. and Kenkre, V.M. (2003). Analytic solutions of a nonlinear convective equation in population dynamics, Physica D 183: 245-259. | Zbl 1059.92043
[019] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Systems, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin/New York, NY. | Zbl 0456.35001
[020] Ichikawa, A. (1982). Quadratic control of evolution equations with delays in control, SIAM Journal Control and Optimization 20(5): 645-668. | Zbl 0495.49006
[021] Kenkre, V.M. (2004). Results from variants of the Fisher equation in the study of epidemics and bacteria, Physica A 342: 242-248.
[022] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, NY, Berlin/Heidelberg/Tokyo. | Zbl 0516.47023
[023] Qaraai, Y. (2008). Compensation des perturbations étalables et contrôle protecteur des zones vulnrables, Ph.D. thesis, Faculty of Sciences and Techniques of Tangier, Morocco.
[024] Qaraai, Y., Bernoussi, A. and El Jai, A. (2006). Contrôle protecteur et remdiabilité, Proceeding of the 3rd International Congress on Scientific Advances in Civil and Industrial Engineering, Algeciras, Spain, pp. 108-114.
[025] Qaraai, Y., Bernoussi, A. and El Jai, A. (2008). How to compensate a spreading disturbance for a class of nonlinear systems, International Journal Applied Mathematics and Computer Science 18(2): 171-187, DOI: 10.2478/v10006008-0016-9. | Zbl 1234.93077
[026] Qaraai, Y., Bernoussi, A. and El Jai, A. (2009). Protector control for a class of nonlinear systems, Proceeding of the International Conference on Systems Theory: Modeling, Analysis and Control, Fes2009, Fes, Morocco, pp. 265-272.
[027] Smart, D.R. (1994). Fixed Point Theorems, Cambridge University Press, London. | Zbl 0427.47036
[028] Zerrik, E., El Jai, A. and Amouroux, M. (1994). Regional observability of distributed systems, International Journal of Systems Science 25(6): 301-313. | Zbl 0812.93015